Program

We will consider the following shadowing properties of discrete dynamical
systems: pseudo-orbit tracing property (or standard shadowing property),
orbital shadowing property and weak shadowing property. We will discuss the
problems of genericity and density of the above-mentioned shadowing
properties in different spaces of discrete dynamical systems on closed
smooth manifolds. Particularly we will talk about nondensity of orbital
shadowing property with respect to C^1-topology.

Irreversibly heading for balance?
Some examples of colorful dynamics in simple diffusion processes

Diffusion equations have been thoroughly studied in the past decades, and
a large variety of results has given rise to some common belief about
effects of diffusion. Typical conclusions of this flavor read as follows.

Bounded solutions stabilize towards some equilibrium.

Diffusive evolution never returns to a state in which it has already
been sometime in the past.

Singularities are melted down either instantaneously or never.

The intention of the talk is to present some exemplary results which
indicate that such dogma-like statements need not be true in general.
To underline this, the main focus will be on cases where the mathematical
setting, made up by the PDE ingredients, space dimensions and domains, is
as simple as possible.

Marek Fila(Comenius University, Bratislava)

Homoclinic and Heteroclinic Orbits for a Semilinear Parabolic Equation

We study the existence of connecting orbits for the Fujita equation,
ut=Δu+up,
with a critical or supercritical exponent p.
For certain ranges of the exponent we prove the existence of heteroclinic connections from
positive steady states to zero and the existence of a homoclinic orbit with respect to zero.
This is a joint work with Eiji Yanagida.